I am afraid the result stated as "well-known" in the problem is false as stated. This sort of thing is a question about Hall-Higman theory, as it is now-known. The 1956 (Proc LMS) Hall-Higman paper was written for a different reason, to do with the Burnside problem, but Theorem B of that paper highlights when failure, relatively rare though it is, can occur. Group theorists, especially those working in solvable groups, but also those working on fusion and transfer, have further developed the theory. Consider the case $G = {\rm SL}(2,3),$, so that $H$ is quaternion of order $8,$ and $C$ is cyclic of order $3$ (and $p =3$). Let $V$ be the natural (2-dimensional absolutely irreducible) module for $G$ over ${\rm GF}(3).$ Then $V$ is faithful and irreducible, but is not free as ${\rm GF}(3)C$-module. A similar example occurs whenever $p = 2^{n}+1$ is a Fermat prime, $C$ is cyclic of order $p$, and $H$ is an extra-special $2$-group of order $2^{2n+1}$ which admits an automorphism $c$ of order $p.$ Then $H$ has a faithful irreducible representation of degree $2^{n}$ over ${\rm GF}(p),$
and this is afforded by a module $V$ which extends to an irreducible module for the semi-direct product $HC,$ where $C = \langle c \rangle$. Since this module has dimension $p-1,$ it is certainly not free as ${\rm GF}(p)C$-module. There are similar examples with $p = 2,$ where $C$ is a cyclic $2$-group, and $H$ is an extra-special $q$-group of order $q(|C|-1)^2$, where $q$ is a Mersenne prime.